Dispersion of particles in an infinite-horizon Lorentz gas

TL;DR

This paper analyzes the particle dispersion in an infinite-horizon Lorentz gas, revealing a universal power-law tail in the distribution and how scatterer arrangement influences the shape, providing an analytical solution for relevant timescales.

Contribution

It offers the first analytical solution for the Lorentz gas kinetics on practical timescales, highlighting the universal tail decay and the impact of scatterer configuration.

Findings

Density tails decay as a universal power law of exponent -3.

The scatterer arrangement influences the distribution shape.

Convergence to Gaussian distribution is ultraslow, making it practically unattainable.

Abstract

We consider a two-dimensional Lorentz gas with infinite horizon. This paradigmatic model consists of pointlike particles undergoing elastic collisions with fixed scatterers arranged on a periodic lattice. It was rigorously shown that when $t\to\infty$, the distribution of particles is Gaussian. However, the convergence to this limit is ultraslow, hence it is practically unattainable. Here we obtain an analytical solution for the Lorentz gas' kinetics on physically relevant timescales, and find that the density in its far tails decays as a universal power law of exponent $-3$. We also show that the arrangement of scatterers is imprinted in the shape of the distribution.